Next we recall some basic facts about Leonard pairs.
Let \(A,A^*\)  denote a Leonard pair on \(V\) .
By the construction, each of \(A\)  and \(A^*\)  is diagonalizable.
By [1]} the eigenspaces of \(A\)  and \(A^*\)  all have dimension one.
Let \(d+1\)  denote the dimension of \(V\) , and
let \(\lbrace \theta _i\rbrace _{i=0}^d\)  denote an ordering of the eigenvalues of \(A\) .
For \(0 \le i \le d\)  let \(v_i\)  denote an eigenvector for \(A\)  corresponding to \(\theta _i\) .
The ordering \(\lbrace \theta _i\rbrace _{i=0}^d\)  is said to be standard whenever the matrix
representing \(A^*\)  with respect to the basis \(\lbrace v_i\rbrace _{i=0}^d\)  is irreducible tridiagonal.
For a standard ordering \(\lbrace \theta _i\rbrace _{i=0}^d\)  of the eigenvalues of \(A\) ,
the ordering \(\lbrace \theta _{d-i}\rbrace _{i=0}^d\)  is standard and no further ordering is
standard.
Similar comments apply to the orderings of the eigenvalues for \(A^*\) .
The Leonard pair \(A,A^*\)  is often described using some data
called a parameter array [2]}.
This is a sequence of scalars
\(( \lbrace \theta _i\rbrace _{i=0}^d; \lbrace \theta ^*_i\rbrace _{i=0}^d; \lbrace \varphi _i\rbrace _{i=1}^d; \lbrace \phi _i\rbrace _{i=1}^d) \) 
